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Mirrors > Home > MPE Home > Th. List > cotr3 | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.) |
Ref | Expression |
---|---|
cotr3.a | ⊢ 𝐴 = dom 𝑅 |
cotr3.b | ⊢ 𝐵 = (𝐴 ∩ 𝐶) |
cotr3.c | ⊢ 𝐶 = ran 𝑅 |
Ref | Expression |
---|---|
cotr3 | ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cotr3.a | . . 3 ⊢ 𝐴 = dom 𝑅 | |
2 | 1 | eqimss2i 4023 | . 2 ⊢ dom 𝑅 ⊆ 𝐴 |
3 | cotr3.b | . . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶) | |
4 | cotr3.c | . . . . 5 ⊢ 𝐶 = ran 𝑅 | |
5 | 1, 4 | ineq12i 4184 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅) |
6 | 3, 5 | eqtri 2841 | . . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅) |
7 | 6 | eqimss2i 4023 | . 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 |
8 | 4 | eqimss2i 4023 | . 2 ⊢ ran 𝑅 ⊆ 𝐶 |
9 | 2, 7, 8 | cotr2 14325 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∀wral 3135 ∩ cin 3932 ⊆ wss 3933 class class class wbr 5057 dom cdm 5548 ran crn 5549 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 |
This theorem is referenced by: (None) |
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